Much cross-sectional data in econometrics is blighted by dependence across units. A solution to this problem is the use of spatial models that allow for an explicit form of dependence across space. This thesis studies problems related to spatial models with increasingly many parameters. A large proportion of the thesis concentrates on Spatial Autoregressive (SAR) models with increasing dimension. Such models are frequently used to model spatial correlation, especially in settings where the data are irregularly spaced. Chapter 1 provides an introduction and background material for the thesis. Chapter 2 develops consistency and asymptotic normality of least squares and instrumental variables (IV) estimates for the parameters of a higher-order spatial autoregressive (SAR) model with regressors. The order of the SAR model and the number of regressors are allowed to approach infinity with sample size, and the permissible rate of growth of the dimension of the parameter space relative to sample size is studied. An alternative to least squares or IV is to use the Gaussian pseudo maximum likelihood estimate (PMLE), studied in Chapter 3. However, this is plagued by finitesample problems due to the implicit definition of the estimate, these being exacerbated by the increasing dimension of the parameter space. A computationally simple Newton type step is used to obtain estimates with the same asymptotic properties as those of the PMLE. Chapters 4 and 5 of the thesis deal with spatial models on an equally spaced, d dimensional lattice. We study the covariance structure of stationary random fields defined on d-dimensional lattices in detail and use the analysis to extend many results from time series. Our main theorem concerns autoregressive spectral density estimation. Stationary random fields on a regularly spaced lattice have an infinite autoregressive representation if they are also purely non-deterministic. We use truncated versions of the AR representation to estimate the spectral density and establish uniform consistency of the proposed spectral density estimate.